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## G = D54order 108 = 22·33

### Dihedral group

Aliases: D54, C2×D27, C54⋊C2, C9.D6, C27⋊C22, C3.D18, C6.2D9, C18.2S3, sometimes denoted D108 or Dih54 or Dih108, SmallGroup(108,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — D54
 Chief series C1 — C3 — C9 — C27 — D27 — D54
 Lower central C27 — D54
 Upper central C1 — C2

Generators and relations for D54
G = < a,b | a54=b2=1, bab=a-1 >

27C2
27C2
27C22
9S3
9S3
9D6
3D9
3D9
3D18

Character table of D54

 class 1 2A 2B 2C 3 6 9A 9B 9C 18A 18B 18C 27A 27B 27C 27D 27E 27F 27G 27H 27I 54A 54B 54C 54D 54E 54F 54G 54H 54I size 1 1 27 27 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ4 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 0 2 2 2 2 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 2 -2 2 2 2 -2 -2 -2 -1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ7 2 -2 0 0 2 -2 -1 -1 -1 1 1 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 orthogonal lifted from D18 ρ8 2 2 0 0 2 2 -1 -1 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ9 2 2 0 0 2 2 -1 -1 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ10 2 2 0 0 2 2 -1 -1 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ11 2 -2 0 0 2 -2 -1 -1 -1 1 1 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 orthogonal lifted from D18 ρ12 2 -2 0 0 2 -2 -1 -1 -1 1 1 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ95-ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ98-ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ97-ζ92 orthogonal lifted from D18 ρ13 2 -2 0 0 -1 1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 ζ2725+ζ272 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2716+ζ2711 -ζ2717-ζ2710 -ζ2716-ζ2711 -ζ2725-ζ272 -ζ2720-ζ277 -ζ2714-ζ2713 -ζ2722-ζ275 -ζ2723-ζ274 -ζ2719-ζ278 -ζ2726-ζ27 orthogonal faithful ρ14 2 2 0 0 -1 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2720+ζ277 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2725+ζ272 ζ2719+ζ278 ζ2725+ζ272 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 orthogonal lifted from D27 ρ15 2 -2 0 0 -1 1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 ζ2722+ζ275 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2714+ζ2713 -ζ2725-ζ272 -ζ2714-ζ2713 -ζ2722-ζ275 -ζ2723-ζ274 -ζ2719-ζ278 -ζ2726-ζ27 -ζ2717-ζ2710 -ζ2720-ζ277 -ζ2716-ζ2711 orthogonal faithful ρ16 2 2 0 0 -1 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2722+ζ275 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2714+ζ2713 ζ2725+ζ272 ζ2714+ζ2713 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 orthogonal lifted from D27 ρ17 2 -2 0 0 -1 1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 ζ2720+ζ277 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2725+ζ272 -ζ2719-ζ278 -ζ2725-ζ272 -ζ2720-ζ277 -ζ2716-ζ2711 -ζ2722-ζ275 -ζ2723-ζ274 -ζ2714-ζ2713 -ζ2726-ζ27 -ζ2717-ζ2710 orthogonal faithful ρ18 2 2 0 0 -1 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2726+ζ27 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2719+ζ278 ζ2722+ζ275 ζ2719+ζ278 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 orthogonal lifted from D27 ρ19 2 -2 0 0 -1 1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 ζ2716+ζ2711 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2720+ζ277 -ζ2726-ζ27 -ζ2720-ζ277 -ζ2716-ζ2711 -ζ2725-ζ272 -ζ2723-ζ274 -ζ2714-ζ2713 -ζ2722-ζ275 -ζ2717-ζ2710 -ζ2719-ζ278 orthogonal faithful ρ20 2 -2 0 0 -1 1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 ζ2726+ζ27 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2717+ζ2710 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2719+ζ278 -ζ2722-ζ275 -ζ2719-ζ278 -ζ2726-ζ27 -ζ2717-ζ2710 -ζ2720-ζ277 -ζ2716-ζ2711 -ζ2725-ζ272 -ζ2723-ζ274 -ζ2714-ζ2713 orthogonal faithful ρ21 2 2 0 0 -1 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2717+ζ2710 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2726+ζ27 ζ2723+ζ274 ζ2726+ζ27 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 orthogonal lifted from D27 ρ22 2 -2 0 0 -1 1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 ζ2719+ζ278 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2717+ζ2710 -ζ2714-ζ2713 -ζ2717-ζ2710 -ζ2719-ζ278 -ζ2726-ζ27 -ζ2725-ζ272 -ζ2720-ζ277 -ζ2716-ζ2711 -ζ2722-ζ275 -ζ2723-ζ274 orthogonal faithful ρ23 2 2 0 0 -1 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2723+ζ274 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2722+ζ275 ζ2720+ζ277 ζ2722+ζ275 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 orthogonal lifted from D27 ρ24 2 -2 0 0 -1 1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 ζ2714+ζ2713 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2723+ζ274 -ζ2716-ζ2711 -ζ2723-ζ274 -ζ2714-ζ2713 -ζ2722-ζ275 -ζ2717-ζ2710 -ζ2719-ζ278 -ζ2726-ζ27 -ζ2725-ζ272 -ζ2720-ζ277 orthogonal faithful ρ25 2 -2 0 0 -1 1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 -ζ2724-ζ273 ζ2717+ζ2710 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2726+ζ27 -ζ2723-ζ274 -ζ2726-ζ27 -ζ2717-ζ2710 -ζ2719-ζ278 -ζ2716-ζ2711 -ζ2725-ζ272 -ζ2720-ζ277 -ζ2714-ζ2713 -ζ2722-ζ275 orthogonal faithful ρ26 2 2 0 0 -1 -1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2714+ζ2713 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2716+ζ2711 ζ2723+ζ274 ζ2716+ζ2711 ζ2723+ζ274 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 orthogonal lifted from D27 ρ27 2 2 0 0 -1 -1 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2719+ζ278 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2717+ζ2710 ζ2714+ζ2713 ζ2717+ζ2710 ζ2719+ζ278 ζ2726+ζ27 ζ2725+ζ272 ζ2720+ζ277 ζ2716+ζ2711 ζ2722+ζ275 ζ2723+ζ274 orthogonal lifted from D27 ρ28 2 -2 0 0 -1 1 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 -ζ2724-ζ273 -ζ2721-ζ276 -ζ2715-ζ2712 ζ2723+ζ274 ζ2719+ζ278 ζ2716+ζ2711 ζ2725+ζ272 ζ2714+ζ2713 ζ2726+ζ27 ζ2717+ζ2710 ζ2720+ζ277 ζ2722+ζ275 -ζ2720-ζ277 -ζ2722-ζ275 -ζ2723-ζ274 -ζ2714-ζ2713 -ζ2726-ζ27 -ζ2717-ζ2710 -ζ2719-ζ278 -ζ2716-ζ2711 -ζ2725-ζ272 orthogonal faithful ρ29 2 2 0 0 -1 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2716+ζ2711 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2726+ζ27 ζ2720+ζ277 ζ2726+ζ27 ζ2720+ζ277 ζ2716+ζ2711 ζ2725+ζ272 ζ2723+ζ274 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2719+ζ278 orthogonal lifted from D27 ρ30 2 2 0 0 -1 -1 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2715+ζ2712 ζ2724+ζ273 ζ2721+ζ276 ζ2725+ζ272 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2717+ζ2710 ζ2716+ζ2711 ζ2717+ζ2710 ζ2716+ζ2711 ζ2725+ζ272 ζ2720+ζ277 ζ2714+ζ2713 ζ2722+ζ275 ζ2723+ζ274 ζ2719+ζ278 ζ2726+ζ27 orthogonal lifted from D27

Smallest permutation representation of D54
On 54 points
Generators in S54
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)```

`G:=sub<Sym(54)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28)])`

D54 is a maximal subgroup of   D108  C27⋊D4  Q8⋊D27
D54 is a maximal quotient of   Dic54  D108  C27⋊D4

Matrix representation of D54 in GL2(𝔽109) generated by

 22 51 58 80
,
 22 51 29 87
`G:=sub<GL(2,GF(109))| [22,58,51,80],[22,29,51,87] >;`

D54 in GAP, Magma, Sage, TeX

`D_{54}`
`% in TeX`

`G:=Group("D54");`
`// GroupNames label`

`G:=SmallGroup(108,4);`
`// by ID`

`G=gap.SmallGroup(108,4);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-3,302,237,1203,138,1804]);`
`// Polycyclic`

`G:=Group<a,b|a^54=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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